Advertisement

Ukrainian Mathematical Journal

, 63:580 | Cite as

Properties of a certain product of submodules

  • M. J. Nikmehr
  • R. Nikandish
  • S. Heidari
Article
  • 62 Downloads

Let R be a commutative ring with identity, let M be an R-module, and let K 1, . . . ,K n be submodules of M: We construct an algebraic object called the product of K 1, . . . ,K n : This structure is equipped with appropriate operations to get an R(M)-module. It is shown that the R(M)-module M n = M . . .M and the R-module M inherit some of the most important properties of each other. Thus, it is shown that M is a projective (flat) R-module if and only if M n is a projective (flat) R(M)-module.

Keywords

Positive Integer Multiplication Module Prime Ideal Local Ring Maximal Ideal 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

References

  1. 1.
    M. M. Ali, “Idealization and theorems of D. D. Anderson,” Commun. Algebra, 34, 4479–4501 (2006).zbMATHCrossRefGoogle Scholar
  2. 2.
    D. D. Anderson and M. Winders, “Idealization of a module,” J. Commutative Algebra, 1, No. 1, 3–56 (2009).MathSciNetzbMATHCrossRefGoogle Scholar
  3. 3.
    D. D. Anderson, “Cancellation modules and related modules,” Lect. Notes Pure Appl. Math., 220, 13–25 (2001).Google Scholar
  4. 4.
    D. D. Anderson, “Some remarks on multiplication ideals,” Math. Jpn., 4, 463–469 (1980).Google Scholar
  5. 5.
    S. E. Atani, “Submodules of multiplication modules,” Taiwan. J. Math., 9, No. 3, 385–396 (2005).zbMATHGoogle Scholar
  6. 6.
    M. F. Atiyah and I. G. Macdonald, Introduction to Commutative Algebra, Addison-Wesley (1969).Google Scholar
  7. 7.
    K. Divaani-Aazar and M. A. Esmkhani, “Associated prime submodules of finitely generated modules,” Commun. Algebra, 33, 4259–4266 (2005).MathSciNetzbMATHCrossRefGoogle Scholar
  8. 8.
    Z. A. El-Bast and P. F. Smith, “Multiplication modules,” Commun. Algebra, 16, 755–799 (1988).MathSciNetzbMATHCrossRefGoogle Scholar
  9. 9.
    C. Faith, Algebra I: Rings, Modules, and Categories, Springer, Berlin (1981).zbMATHGoogle Scholar
  10. 10.
    J. A. Huckaba, Commutative Rings with Zero Divisors, Marcel Dekker, New York (1988).zbMATHGoogle Scholar
  11. 11.
    H. Matsumara, Commutative Ring Theory, Cambridge University Press, Cambridge (1986).Google Scholar
  12. 12.
    M. Nagata, Local Rings, Interscience, New York (1962).zbMATHGoogle Scholar
  13. 13.
    A. G. Naoum and A. S. Mijbas, “Weak cancellation modules,” Kyungpook Math. J., 37, 73–82 (1997).MathSciNetzbMATHGoogle Scholar

Copyright information

© Springer Science+Business Media, Inc. 2011

Authors and Affiliations

  • M. J. Nikmehr
    • 1
  • R. Nikandish
    • 1
  • S. Heidari
    • 1
  1. 1.Toosi University of TechnologyTehranIran

Personalised recommendations